Ansolabehere and Konisky (2006) want to explain voter turnout Yi,t in county i and yeart. Let Xi,t be 1 if county i in yeart required registration before voting, else 0; let Zi,t be a 1×p vector of control variables. The authors consider two regression models. The first is

(24) Yi,t = α + βXi,t + Zi,tγ + δi,t where δi,t is a random error term. The second is obtained by taking differences:

(25) Yi,t − Yi,t−1 = β(Xi,t − Xi,t−1) + (Zi,t − Zi,t−1)γ + $i,t where $i,t is a random error term. The chief interest is in β, whereas γ is p×1 vector of nuisance parameters. If (24)satisfies the usual conditions for an OLS regression model, what about (25)? And vice versa?